American Institute of Mathematical Sciences

Advanced
June  2007, 7(4): 779-792. doi: 10.3934/dcdsb.2007.7.779

Circular and elliptic orbits in a feedback-mediated chemostat

Willard S. Keeran 1, , Patrick D. Leenheer 1, and  Sergei S. Pilyugin 1,

1. 

Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, United States, United States, United States

Received  August 2006 Revised  January 2007 Published  March 2007

A chemostat with two organisms competing for a single growth-limiting nutrient controlled by feedback-mediated dilution rate is analyzed. A specific feedback function is constructed which yields circular and elliptical periodic orbits for the limiting system. A theorem on the stabilization of periodic orbits in conservative systems is developed and for a given elliptical orbit, the result is used to modify the chemostat so that the chosen orbit is asymptotically stable. Finally, the feedback function is modified so that finitely many nested periodic orbits of alternating stability exist.
Keywords: conservative system., feedback-mediated control, periodic coexistence, Chemostat.
Mathematics Subject Classification: 92D25, 92D37, 92D4.
Citation: Willard S. Keeran, Patrick D. Leenheer, Sergei S. Pilyugin. Circular and elliptic orbits in a feedback-mediated chemostat. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 779-792. doi: 10.3934/dcdsb.2007.7.779
[1]

Willard S. Keeran, Patrick D. Leenheer, Sergei S. Pilyugin. Feedback-mediated coexistence and oscillations in the chemostat. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 321-351. doi: 10.3934/dcdsb.2008.9.321
[2]

Charlotte Beauthier, Joseph J. Winkin, Denis Dochain. Input/state invariant LQ-optimal control: Application to competitive coexistence in a chemostat. Evolution Equations & Control Theory, 2015, 4 (2) : 143-158. doi: 10.3934/eect.2015.4.143
[3]

Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164
[4]

Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087
[5]

Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari. Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences & Engineering, 2016, 13 (4) : 631-652. doi: 10.3934/mbe.2016012
[6]

Gonzalo Robledo. Feedback stabilization for a chemostat with delayed output. Mathematical Biosciences & Engineering, 2009, 6 (3) : 629-647. doi: 10.3934/mbe.2009.6.629
[7]

Evan C. Haskell, Jonathan Bell. Pattern formation in a predator-mediated coexistence model with prey-taxis. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2895-2921. doi: 10.3934/dcdsb.2020045
[8]

Azmy S. Ackleh, Youssef M. Dib, S. R.-J. Jang. Competitive exclusion and coexistence in a nonlinear refuge-mediated selection model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 683-698. doi: 10.3934/dcdsb.2007.7.683
[9]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079
[10]

Richard L Buckalew. Cell cycle clustering and quorum sensing in a response / signaling mediated feedback model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 867-881. doi: 10.3934/dcdsb.2014.19.867
[11]

Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer. On the stability of periodic solutions in the perturbed chemostat. Mathematical Biosciences & Engineering, 2007, 4 (2) : 319-338. doi: 10.3934/mbe.2007.4.319
[12]

Ran Dong, Xuerong Mao. Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations. Mathematical Control & Related Fields, 2020, 10 (4) : 715-734. doi: 10.3934/mcrf.2020017
[13]

Shikun Wang. Dynamics of a chemostat system with two patches. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6261-6278. doi: 10.3934/dcdsb.2019138
[14]

Isabel Coelho, Carlota Rebelo, Elisa Sovrano. Extinction or coexistence in periodic Kolmogorov systems of competitive type. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5743-5764. doi: 10.3934/dcds.2021094
[15]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83
[16]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59
[17]

Corrado Falcolini, Laura Tedeschini-Lalli. A numerical renormalization method for quasi–conservative periodic attractors. Journal of Computational Dynamics, 2020, 7 (2) : 461-468. doi: 10.3934/jcd.2020018
[18]

Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443
[19]

Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 2. Periodic solutions. Conference Publications, 2011, 2011 (Special) : 941-952. doi: 10.3934/proc.2011.2011.941
[20]

Ningning Ye, Zengyun Hu, Zhidong Teng. Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2022022

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy